On critical points of functionals with polyconvex integrands

Let Omega subset of R-n be a bounded domain with Lipschitz boundary, and assume that f : Omega x R-mxn --> R-. is a Caratheodory integrand such that f (x, (.)) is polyconvex for L-n- a.e. x is an element of Omega. In this paper we consider integral functionals of the form F(u, Omega) := integral(Omega) f(x, Du(x)) dx, where f satisfies a growth condition of the type \f(x, A)\ less than or equal to c(1 + \A\(P)), for some c > 0 and 1 < p < infinity, and u lies in the Sobolev space of vector-valued functions W-1,W-p (Omega, R-m). We study the implications of a function u(0) being a critical point of F. In this regard we show among other things that if f does not depend on the spatial variable x, then every piecewise affine critical point of T is a global minimizer subject to its own boundary condition. Moreover for the general case, we construct an example exhibiting that the uniform positivity of the second variation at a critical point is not sufficient for it to be a strong local minimizer. In this example f is discontinuous in x but smooth in A.